Assignment

This week’s assignment is about a signaling game.

• First Nature reveals a type (A or B);
• Then Proposer sends a signal (Left or Right);
• Then Responder, seeing the signal but not the state, chooses and action (Up or Down).

Figure 1: Tree for Weekly 5 practice version

Strategies

In this tree, Proposer has four possible strategies:

1. Left if A, Left if B (LL)
2. Left if A, Right if B (LR)
3. Right if A, Left if B (RL)
4. Right if A, Right if B (RR)

Strategies

And Responder has four possible strategies

1. Up if Left, Up if Right (UU)
2. Up if Left, Down if Right (UD)
3. Down if Left, Up if Right (DU)
4. Down if Left, Down if Right (DD)

Strategy Pairs

So the strategy table has four rows for Proposer, and four columns for Responder, and that means 16 cells.

If there was no randomness, then for any pair, you could tell what each player would get.

But there is randomness. So what do we do?

Answers

I’ll start with the answers, then work back to how to figure them out.

Bottom Right

Let’s start in the bottom right, the 0.6, 2.2.

The formula for P1’s expected return is:

• Probability of A times P1’s return if A happens (and they play these strategies); plus
• Probability of B times P1’s return if B happens (and they play these strategies)

Bottom Right

Let’s start in the bottom right, the 0.6, 2.2.

The formula for P2’s expected return is:

• Probability of A times P2’s return if A happens (and they play these strategies); plus
• Probability of B times P2’s return if B happens (and they play these strategies)

Bottom Right

If A happens, and P1 plays RR, that means P1 chooses R (the first letter in P1’s strategy).

If P1 chooses R, and P2 plays UU, that means P2 chooses U (the second letter in P1’s strategy).

So the row of the table we’re on is where A, then R, then U happen.

And at that row P1 gets 1, and P2 gets 1.

Bottom Right

If B happens, and P1 plays RR, that means P1 chooses R (the first letter in P1’s strategy).

If P1 chooses R, and P2 plays UU, that means P2 chooses U (the second letter in P1’s strategy).

So the row of the table we’re on is where B, then R, then U happen.

And at that row P1 gets 0, and P2 gets 4.

The Formula

So P1’s expected return is:

• 0.6 times 1, plus 0.4 times 0
• which is 0.6.

And P2’s expected return is:

• 0.6 times 1 plus 0.4 times 4
• which is 2.2

RL/UD

Let’s do one more of these: the one where Player 1 plays RL and Player 2 plays UD.

The formulae are still the same, but we have to be a bit more careful about the applications.

RL/UD

As the table says, the results are 3.6 and 1.0. Let’s work to get there.

The formula for P1’s expected return is:

• Probability of A times P1’s return if A happens (and they play these strategies); plus
• Probability of B times P1’s return if B happens (and they play these strategies)

RL/UD

If A happens, and P1 plays RL, that means P1 chooses R (the first letter in P1’s strategy).

If P1 chooses R, and P2 plays UD, that means P2 chooses D (the second letter in P1’s strategy).

So the row of the table we’re on is where A, then R, then D happen.

And at that row P1 gets 4, and P2 gets 1.

RL/UD

If B happens, and P1 plays RL, that means P1 chooses L (the second letter in P1’s strategy).

If P1 chooses L, and P2 plays UD, that means P2 chooses U (the first letter in P1’s strategy).

So the row of the table we’re on is where B, then L, then U happen.

And at that row P1 gets 3, and P2 gets 1.

The Formula

So P1’s expected return is:

• 0.6 times 4, plus 0.4 times 3
• which is 3.6.

And P2’s expected return is

• 0.6 times 1, plus 0.4 times 1
• which is 1

Equilibria

Now we have to find the equilibria for the game.

An equilibria is a pair of choices where neither party can do better by changing their view.

So the top right corner: LL, UU is an equilibrium.

Equilibria

If P1 plays LL, then P2 gets (in expectation)

• 0 if they play DD;
• 0 if they play DU;
• 1 if they play UD;
• 1 if they play UU.

They can’t do better than playing UU. (Ties aren’t a problem.)

Equilibria

If P2 plays UU, then P1 gets (in expectation)

• 2.4 if they play LL;
• 1.2 if they play LR;
• 1.8 if they play RL;
• 0.6 if they play RR.

They can’t do better than playing LL.

Finding Equilibria

We don’t want to have to check all 16 cells this way. Fortunately, we can speed things up.

There are a few approaches - here’s one.

Finding Equilibria

Go through each column, and find the highest score for P1.

That will be P1’s best response to that strategy for P2.

Note that there may be more than one of these, if there are ties.

Finding Equilibria

Then for each of the things you find, check whether P2 can do better by switching.

If so, it’s not an equilibria.

If not, it is one.

And you’ll find all the equilibria this way.

Equilibria

Applying this technique shows that LL,UU and RL,UD are the only equilibria.

Pooling/Separating

An equilibrium is pooling if P1 does the same thing in each state. So P1 plays LL or RR.

An equilibrium is separating if P1 does different things when the states are different. So P2 plays LR or RL.

Further Questions

If we were doing more game theory, there are more questions we could ask at this point.

Some equilibria that you can find this way are not intuitively sensible.

Further Questions

Sometimes that’s because they imply P2 will do things that no longer make sense once they see P1’s move. The notion of sequential equilibrium was developed to rule out these.

For real life cases, there are times when it is sensible to bluff, but not sensible to stick with the bluff once it has been exposed. But the procedure we’ve outlined here might include equilibria where committing to the bluff even when the cards are shown might be an equilibrium.

Further Questions

And into the really not on the exam stuff, there is the question of what to do with the equilibrium in the US-UK game where P1 believes that P2 believes that P1 will buy tea if and only if American.

It is somewhat less clear that this creats real life problems.